Truth vs. mathematical proof

[Vort]

What is the general understanding of a "mathematical proof"?

 

For some years now, I have been engaged in an ongoing conversation regarding the "blue forehead room" problem*. At one point, I offered a mathematical proof by induction of the correct answer. (Remember that this is in a forum with primarily mathematically inclined people.) I naively thought my answer would resolve the matter and put an end to further discussion. Not so. On the contrary, people actually said to me (in effect), "Nice mathematical proof, but in the REAL world, it's not that easy."

 

What do people think a "mathematical proof" is? Something like a "nice idea" or a "clever suggestion"?

 

If someone showed you a mathematical proof of something, and you accepted the proof as valid, would you then believe his answer to be true? Or would you insist that maybe, you know, something else might be the case, regardless of the supposed "proof" that you acknowledge is valid?

 

Serious question. I really wonder what people think "mathematical proof" means.

 

*The problem can be given as follows:

 

Ten perfect logicians, each with a blue mark on his/her forehead, are placed in a room with the following instruction:

 

At least one person has a blue mark on his/her forehead. When you know whether or not you have a blue mark on your forehead, leave the room when the lights go out.

 

Each day, once per day, the lights go out. On which day, if any, do the logicians leave the room? (Caveat: The logicians are not allowed to communicate with each other or use mirrors or other physical means to determine whether they have a spot on their forehead. They may only observe the actions of the others.)

 

SPOILER: The answer is:

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When the lights go out on the tenth day, everyone leaves.

[MrShorty]

If someone showed you a mathematical proof of something, and you accepted the proof as valid, would you then believe his answer to be true? Or would you insist that maybe, you know, something else might be the case, regardless of the supposed "proof" that you acknowledge is valid?

 

I am reminded of my freshman year in high school. Our algebra text presented one of those "proof that 2=1" problems. At first glance, their proof looked correct, but I could not bring myself to believe that 2=1. So I spent some time with the problem until I found the flaw in the logic. If you think someone's proof is wrong, it takes more than a "that can't be true" to negate the proof. You need to spend enough time and effort on the logic to find the flaw(s).

 

FWIW, this also leads me to remember Alice's conversation about unbirthdays with Humpty Dumpty. To paraphrase, Alice explains that 365-1=364, HD says that it sounds right, but he will need to see the sum worked on paper, so Alice writes it out for him. He agrees that it appears to have been done correctly, but he doesn't have the time or inclination to think it through completely. Since inductive reasoning can be among the more difficult concepts to grasp, I wonder if the "that's good in theory but doesn't work in practice" is code for "I don't fully understand inductive reasoning, and, therefore, cannot find the flaw in your logic, but there must be one."

 

To sum up, if someone presents a proof that I agree is valid, then I have to accept the conclusion. If I don't like the conclusion, then it is my job to find the flaw in the logic or to provide a "counter proof".

 

Not sure if that helps at all. If nothing else, we got to remember Alice in her quest to be a queen, and that has to be worth something.

[Traveler]

What is the general understanding of a "mathematical proof"?

 

For some years now, I have been engaged in an ongoing conversation regarding the "blue forehead room" problem*. At one point, I offered a mathematical proof by induction of the correct answer. (Remember that this is in a forum with primarily mathematically inclined people.) I naively thought my answer would resolve the matter and put an end to further discussion. Not so. On the contrary, people actually said to me (in effect), "Nice mathematical proof, but in the REAL world, it's not that easy."

 

What do people think a "mathematical proof" is? Something like a "nice idea" or a "clever suggestion"?

 

If someone showed you a mathematical proof of something, and you accepted the proof as valid, would you then believe his answer to be true? Or would you insist that maybe, you know, something else might be the case, regardless of the supposed "proof" that you acknowledge is valid?

 

Serious question. I really wonder what people think "mathematical proof" means.

 

*The problem can be given as follows:

 

Ten perfect logicians, each with a blue mark on his/her forehead, are placed in a room with the following instruction:

 

At least one person has a blue mark on his/her forehead. When you know whether or not you have a blue mark on your forehead, leave the room when the lights go out.

 

Each day, once per day, the lights go out. On which day, if any, do the logicians leave the room? (Caveat: The logicians are not allowed to communicate with each other or use mirrors or other physical means to determine whether they have a spot on their forehead. They may only observe the actions of the others.)

 

SPOILER: The answer is:

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When the lights go out on the tenth day, everyone leaves.

 

You and your critics are both right - there are no true perfect logicians in the real world.  If there were - they would all leave on the 10th day.  Those that argue against this are only displaying their own logical breakdowns.  It is kind of like my brother's favorite statement about real world logicians.  "There are 3 kinds - those that can do math and those that can't."

Edited April 27, 2015 by Traveler

[The Folk Prophet]

When the lights go out on the tenth day, everyone leaves.

 

I would like for you to show your work.

[mordorbund]

What is the general understanding of a "mathematical proof"?

 

For some years now, I have been engaged in an ongoing conversation regarding the "blue forehead room" problem*. At one point, I offered a mathematical proof by induction of the correct answer. (Remember that this is in a forum with primarily mathematically inclined people.) I naively thought my answer would resolve the matter and put an end to further discussion. Not so. On the contrary, people actually said to me (in effect), "Nice mathematical proof, but in the REAL world, it's not that easy."

 

What do people think a "mathematical proof" is? Something like a "nice idea" or a "clever suggestion"?

 

If someone showed you a mathematical proof of something, and you accepted the proof as valid, would you then believe his answer to be true? Or would you insist that maybe, you know, something else might be the case, regardless of the supposed "proof" that you acknowledge is valid?

 

Serious question. I really wonder what people think "mathematical proof" means.

 

*The problem can be given as follows:

 

Ten perfect logicians, each with a blue mark on his/her forehead, are placed in a room with the following instruction:

 

At least one person has a blue mark on his/her forehead. When you know whether or not you have a blue mark on your forehead, leave the room when the lights go out.

 

Each day, once per day, the lights go out. On which day, if any, do the logicians leave the room? (Caveat: The logicians are not allowed to communicate with each other or use mirrors or other physical means to determine whether they have a spot on their forehead. They may only observe the actions of the others.)

 

SPOILER: The answer is:

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.

.

.

.

.

.

.

.

.

.

.

.

.

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.

.

.

.

.

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When the lights go out on the tenth day, everyone leaves.

 

Math is useful for creating models. Insofar as the model is an accurate representation of "the real world" the associated proofs are useful. Take the old joke about lining the boys up on one side of the gym and the girls on the other. Each iteration they reduce the distance by half. They will approach contact but never reach it. The joke is that this is the conclusion the mathematician reaches, but the engineer says they just need to be "close enough for practical purposes". Well, you've just changed the question haven't you. It's no longer "when will they contact" but "when will they be close enough" with built-in assumptions about whether arms reach out after closing the distance or if that's included in the calculation. At the end of the day, you've just changed the scenario that the model was built for. If the scenario took place in "the real world", then that reality was not understood well enough to account for it in the model.

[mordorbund]

I would like for you to show your work.

 

He told you how it works. Via induction:

 

1. On day 1 the white coat says "Not only is at least one of you in the blue-forehead frat, but you all should want to be in the BFF. It's pretty rad."
2. On day 2 the logicians are left to gossip amongst themselves all the rumors they've heard about the blue-forehead frat and how prestigious it is.
3. On day 3 the white coat enters again and wonders aloud if this group has what it takes.
4. On day 4 the white coat says he might just bring a mirror in the room and shame them all, but he has too much respect for BFF to go against tradition.
5. On day 5 they have to run laps. Just because.
6. On day 6 the white coat says that they should start considering who is the weakest logician amongst them.
7. On day 7 several white coats come in with false premises trying to trick the initiates into talking.
8. On day 8 they knit hats for the local cancer ward (what - logicians can't try to get into service fraternities?).
9. On day 9 the white coats say they'll have to start rationing food if they can't get their act together.
10. On day 10 the initiates struggle to contain their joy as they wait for the lights to go out again.
11. They have a party welcoming the new BFF members.

[Vort]

I would like for you to show your work.

 

'k.

 

Proof by induction that a claim is true for all n works as follows:

1. Show that the claim is true for n=1 (or n=0, or some other initial condition).
2. Assume that the claim is true for some number m, then show that it must also be true for m + 1.

If the above two conditions are met, then you can generalize (by induction) that the claim is true for all n. This is true because, by condition (1), you have shown the claim is true for 1. Therefore, by condition (2), you have shown it is true for 1+1=2. But since it's true for 2, then by condition (2), it must also be true for 2+1=3, and therefore for 3+1=4, and so on for every number >1.

 

1. Show that the claim is true for n = 1.

 

Suppose we have ten (or any other number of) logicians, one of whom has a blue forehead and the others not. When they are placed in the room and given instructions, the one with the blue forehead looks at his companions and thinks, "At least one of us has a blue forehead. None of my companions has a blue forehead. Therefore, I must have the blue forehead." Thus, when the lights go out, the person with the blue forehead leaves. That's after ONE day, same as the number of blue foreheads. So the claim is true for n = 1.

 

(Note that those without a blue forehead do not know whether or not their own forehead is blue. All they know is that the other guy has a blue forehead, so the minimal condition is fulfilled. But that does not tell them whether their own forehead is blue.)

 

2. Assume that the claim is true for some number m, then show that it must also be true for m + 1.

 

So let's say that some number m of the logicians have blue foreheads (let m be greater than zero, as the problem requires, and less than the number of logicians), and that we know that it requires m days for them (the blue-foreheaded people) to realize their foreheads are blue. This is our assumption.

 

Let's now posit there are m+1 blue-foreheaded logicians in the room. Each of them will count m blue foreheads (all except their own) and realize that, as we have assumed above, all the blue-foreheaded people will leave after m days -- unless their own forehead is also blue. In that case, after the lights go out on the 30th (or whatever number m is) day and then come back on, they will observe that none of the other 30 blue-foreheaded people left.

 

This is precisely what happens. Thus, they will deduce that their own forehead must be blue, and when the lights go out on the m+1th day, they (and the rest of the blue-foreheaded people -- m+1 in all) will leave.

 

Thus, if n perfect logicians have blue foreheads, they will all leave on the nth day. QED.

Edited April 27, 2015 by Vort

[mordorbund]

1. Assume you have 1 out of 1 with the blue mark. That person constitutes "at least 1" and would leave once the lights went out (it's the same for 1/10 - since the 1 would not see anyone with the mark).

2. Assume you have 2 out of 2 with the blue mark (A and B). A would see B has a blue mark, so B should leave when the lights went out. B doesn't leave, so A knows that B has some ambiguity (the same ambiguity A has). So A knows they both have the blue mark and leaves the next time the lights go out.

3. Assume this hold true for k out of k with the blue mark. All k of them leave on the kth iteration.

4. For k+1, (K+1) looks around and sees all K of them have the blue mark, so K expects them all to leave on the kth iteration (see 3 and the explanation for 2). When they do not leave, that tells (K+1) that (K+1) also has a blue mark and they all leave on the Kth+1 iteration.

 

1 is the base case. 2 explains better why K+1 works. 4 clinches it.

[The Folk Prophet]

Or, in other words the deduction is based on the deductions (or lack thereof) of the others in the room.

 

 

Interesting.

[Crypto]

Change the basic axioms and any proof can be invalidated.

[Vort]

Change the basic axioms and any proof can be invalidated.

 

Not sure I understand your meaning, Crypto. Can you give me an example of what you're saying? For example, which basic axioms would you change to invalidate my proof? You mean like the logical axiom (A) and (B) = (A and B)?

[Crypto]

I'm not a mathematician, so I only have a basic understanding of axioms, but if the axioms on which math are based were changed, Logical rules on which the math function would not apply.

For example, if a number is not equal to itself A=/=A, then a proof could in theory no longer be accepted.

Though someone arguing as such, is probably on weak ground to begin with.

Edited April 28, 2015 by Crypto

[mordorbund]

For example, if a number is not equal to itself A=/=A, then a proof could in theory no longer be accepted.

 

 

Nah. Logicians use that tool for proof by contradiction :)

[mordorbund]

Or, in other words the deduction is based on the deductions (or lack thereof) of the others in the room.

 

 

Interesting.

 

You can do that when dealing with "perfect logicians". Like the physicist's spherical cow.

[Guest]

The difference between mathematical proof and real life is simple. In real life, there are no perfect logicians.

[mordorbund]

The difference between mathematical proof and real life is simple. In real life, there are no perfect logicians.

 

Or, in other words the deduction is based on the deductions (or lack thereof) of the others in the room.

 

 

Interesting.

 

This could be fun. Let's assume that n out of the 10 are doofuses. They may or may not have blue marks on their heads. Is there a point where you (as a purported perfect logician) can ring the bell and declare whether you have a blue mark? Even with the uncertainty of how many doofuses there are?

[Guest]

This could be fun. Let's assume that n out of the 10 are doofuses. They may or may not have blue marks on their heads. Is there a point where you (as a purported perfect logician) can ring the bell and declare whether you have a blue mark? Even with the uncertainty of how many doofuses there are?

The perfect logicians will be left standing at the 100th iteration as they don't have enough reliable input to deduce knowledge. The doofuses, of course, may or may not have left the room.

But... I made a mistake in that statement. "There are no perfect logicians" is an illogical statement. Rather, I should have said, "In real life, logicians can't assume all the other logicians are perfect logicians".

Edited April 28, 2015 by anatess

[Traveler]

'k.

 

Proof by induction that a claim is true for all n works as follows:

1. Show that the claim is true for n=1 (or n=0, or some other initial condition).
2. Assume that the claim is true for some number m, then show that it must also be true for m + 1.

If the above two conditions are met, then you can generalize (by induction) that the claim is true for all n. This is true because, by condition (1), you have shown the claim is true for 1. Therefore, by condition (2), you have shown it is true for 1+1=2. But since it's true for 2, then by condition (2), it must also be true for 2+1=3, and therefore for 3+1=4, and so on for every number >1.

 

1. Show that the claim is true for n = 1.

 

Suppose we have ten (or any other number of) logicians, one of whom has a blue forehead and the others not. When they are placed in the room and given instructions, the one with the blue forehead looks at his companions and thinks, "At least one of us has a blue forehead. None of my companions has a blue forehead. Therefore, I must have the blue forehead." Thus, when the lights go out, the person with the blue forehead leaves. That's after ONE day, same as the number of blue foreheads. So the claim is true for n = 1.

 

(Note that those without a blue forehead do not know whether or not their own forehead is blue. All they know is that the other guy has a blue forehead, so the minimal condition is fulfilled. But that does not tell them whether their own forehead is blue.)

 

2. Assume that the claim is true for some number m, then show that it must also be true for m + 1.

 

So let's say that some number m of the logicians have blue foreheads (let m be greater than zero, as the problem requires, and less than the number of logicians), and that we know that it requires m days for them (the blue-foreheaded people) to realize their foreheads are blue. This is our assumption.

 

Let's now posit there are m+1 blue-foreheaded logicians in the room. Each of them will count m blue foreheads (all except their own) and realize that, as we have assumed above, all the blue-foreheaded people will leave after m days -- unless their own forehead is also blue. In that case, after the lights go out on the 30th (or whatever number m is) day and then come back on, they will observe that none of the other 30 blue-foreheaded people left.

 

This is precisely what happens. Thus, they will deduce that their own forehead must be blue, and when the lights go out on the m+1th day, they (and the rest of the blue-foreheaded people -- m+1 in all) will leave.

 

Thus, if n perfect logicians have blue foreheads, they will all leave on the nth day. QED.

 

Vort - your proof is rhetorically valid.  The opposition or discussion outside of what you have provided only proves those so involved do not know what they are talking about.

[Guest]

Vort - your proof is rhetorically valid.  The opposition or discussion outside of what you have provided only proves those so involved do not know what they are talking about.

Disagree.

From Vort's OP, the issue is that some people do not accept Vort's mathematical solution as proof applicable to real life.

Vort's proof may be rhetorically valid but it is based on an assumption. The opposition does not prove - in any way shape or form - that they don't know what they are talking about. The opposition merely rejects the assumption that the mathematical proof is based on as applicable to real life.

[Vort]

Vort's proof may be rhetorically valid but it is based on an assumption. The opposition does not prove - in any way shape or form - that they don't know what they are talking about. The opposition merely rejects the assumption that the mathematical proof is based on as applicable to real life.

 

What assumption is my proof based on that can be invalidated?

[mordorbund]

The perfect logicians will be left standing at the 100th iteration as they don't have enough reliable input to deduce knowledge. The doofuses, of course, may or may not have left the room.

But... I made a mistake in that statement. "There are no perfect logicians" is an illogical statement. Rather, I should have said, "In real life, logicians can't assume all the other logicians are perfect logicians".

 

Baloney. I've worked it out informally, and the only case where the logician is still left in the room is if there is exactly 0 blue marks, or 1 and the logician is that one. Whenever that happens, the experiment needs to get reset by a white coat stating "1 or more of you has a blue head". With that, the only way the logician is left in the lurch is if the logician is not told about the reset rule.

[Vort]

It occurs to me that my OP might not be clear.

 

When I offered my mathematical proof and thought to end all contrary discussion, I was told, in effect, that in the "real world" it wasn't so easy. In this case, the "real world" did not mean "a world where 100 perfect logicians do not exist" or some such. Rather, the intent was that a proof may work on paper, but in an actual situation AS DESCRIBED, it was not workable.

 

That is, if you actually put ten (or however many) "perfect logicians" in a room with the stated rules and under the stated conditions -- "perfect" enough to be able to reason out the problem -- they could not use my stated reasoning to arrive independently at the same conclusion. Rather, there would be confounding factors.

 

It was this insistence that I was marvelling at, and wondering what people thought a "mathematical proof" meant. There was no idea that, you know, people aren't perfect, and most people couldn't reason this out, blah, blah. That wasn't at all the point. The point was that, if people COULD and DID reason it out, it would still be wrong. (According to my critics.)

[Guest]

What assumption is my proof based on that can be invalidated?

The assumption that all 100 people are perfect logicians. You can't assume that in real life even if the test giver establishes that they are all perfect logicians.

Therefore, when you're programming this stuff, you have to allow for error (logicians come to the wrong conclusion) or put disclaimers all over the place.

And that's why... the mathematical model for Climate Change can't really be used to prove Climate Change is a bigger concern than ISIS.

Edited April 28, 2015 by anatess

[mordorbund]

It occurs to me that my OP might not be clear.

 

When I offered my mathematical proof and thought to end all contrary discussion, I was told, in effect, that in the "real world" it wasn't so easy. In this case, the "real world" did not mean "a world where 100 perfect logicians do not exist" or some such. Rather, the intent was that a proof may work on paper, but in an actual situation AS DESCRIBED, it was not workable.

 

That is, if you actually put ten (or however many) "perfect logicians" in a room with the stated rules and under the stated conditions -- "perfect" enough to be able to reason out the problem -- they could not use my stated reasoning to arrive independently at the same conclusion. Rather, there would be confounding factors.

 

It was this insistence that I was marvelling at, and wondering what people thought a "mathematical proof" meant. There was no idea that, you know, people aren't perfect, and most people couldn't reason this out, blah, blah. That wasn't at all the point. The point was that, if people COULD and DID reason it out, it would still be wrong. (According to my critics.)

 

To your critics, your avatar serves as my rebuttal.

[Guest]

It occurs to me that my OP might not be clear.

 

When I offered my mathematical proof and thought to end all contrary discussion, I was told, in effect, that in the "real world" it wasn't so easy. In this case, the "real world" did not mean "a world where 100 perfect logicians do not exist" or some such. Rather, the intent was that a proof may work on paper, but in an actual situation AS DESCRIBED, it was not workable.

 

That is, if you actually put ten (or however many) "perfect logicians" in a room with the stated rules and under the stated conditions -- "perfect" enough to be able to reason out the problem -- they could not use my stated reasoning to arrive independently at the same conclusion. Rather, there would be confounding factors.

 

It was this insistence that I was marvelling at, and wondering what people thought a "mathematical proof" meant. There was no idea that, you know, people aren't perfect, and most people couldn't reason this out, blah, blah. That wasn't at all the point. The point was that, if people COULD and DID reason it out, it would still be wrong. (According to my critics.)

By logic, as you had this discussion with mathematicians, there are 2 conclusions... that 1.) these mathematicians are doofuses or 2.) you misunderstood their sentiments when all they really wanted to say was that they do not trust the assumption of perfect logicians (as is applicable in the real world) so they have to account for error.